This document discusses extensions to the Modigliani-Miller (MM) capital structure models. It covers MM models with corporate taxes, the Miller model with personal taxes, growth assumptions, risky debt, and values equity as a call option when debt is risky. Key points include that MM models predict firm value increases with leverage due to tax benefits, while Miller's model shows a smaller benefit due to personal taxes favoring equity. Growth assumptions change the value and discount rate of tax shields. Risky debt can value equity as a call option on the firm's assets.

1. 17 - 1
CHAPTER 17
Capital Structure Decisions:
Extensions
MM and Miller models
Hamada’s equation
Financial distress and agency costs
Trade-off models
Asymmetric information theory

2. 17 - 2
Who are Modigliani and Miller (MM)?
They published theoretical papers
that changed the way people thought
about financial leverage.
They won Nobel prizes in economics
because of their work.
MM’s papers were published in 1958
and 1963. Miller had a separate
paper in 1977. The papers differed in
their assumptions about taxes.

3. 17 - 3
What assumptions underlie the MM
and Miller models?
Firms can be grouped into
homogeneous classes based on
business risk.
Investors have identical
expectations about firms’ future
earnings.
There are no transactions costs.
(More...)

4. 17 - 4
All debt is riskless, and both
individuals and corporations can
borrow unlimited amounts of money
at the risk-free rate.
All cash flows are perpetuities. This
implies perpetual debt is issued,
firms have zero growth, and
expected EBIT is constant over time.
(More...)

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MM’s first paper (1958) assumed
zero taxes. Later papers added
taxes.
No agency or financial distress
costs.
These assumptions were necessary
for MM to prove their propositions
on the basis of investor arbitrage.

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Proposition I:
VL = VU.
Proposition II:
rsL = rsU + (rsU - rd)(D/S).
MM with Zero Taxes (1958)

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Firms U and L are in same risk class.
EBITU,L = $500,000.
Firm U has no debt; rsU = 14%.
Firm L has $1,000,000 debt at rd = 8%.
The basic MM assumptions hold.
There are no corporate or personal taxes.
Given the following data, find V, S,
rs, and WACC for Firms U and L.

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1. Find VU and VL.
VU = = = $3,571,429.
VL = VU = $3,571,429.
EBIT
rsU
$500,000
0.14

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VL = D + S = $3,571,429
$3,571,429 = $1,000,000 + S
S = $2,571,429.
2. Find the market value of
Firm L’s debt and equity.

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3. Find rsL.
rsL = rsU + (rsU - rd)(D/S)
= 14.0% + (14.0% - 8.0%)( )
= 14.0% + 2.33% = 16.33%.
$1,000,000
$2,571,429

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4. Proposition I implies WACC = rsU.
Verify for L using WACC formula.
WACC = wdrd + wcers = (D/V)rd + (S/V)rs
= ( )(8.0%)
+( )(16.33%)
= 2.24% + 11.76% = 14.00%.
$1,000,000
$3,571,429
$2,571,429
$3,571,429

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Graph the MM relationships between
capital costs and leverage as measured
by D/V.
Without taxesCost of
Capital (%)
26
20
14
8
0 20 40 60 80 100
Debt/Value
Ratio (%)
rs
WACC
rd

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The more debt the firm adds to its
capital structure, the riskier the
equity becomes and thus the higher
its cost.
Although rd remains constant, rs
increases with leverage. The
increase in rs is exactly sufficient to
keep the WACC constant.

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Graph value versus leverage.
Value of Firm, V (%)
4
3
2
1
0 0.5 1.0 1.5 2.0 2.5
Debt (millions of $)
VLVU
Firm value ($3.6 million)
With zero taxes, MM argue that value
is unaffected by leverage.

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Find V, S, rs, and WACC for Firms U and
L assuming a 40% corporate
tax rate.
With corporate taxes added, the MM
propositions become:
Proposition I:
VL = VU + TD.
Proposition II:
rsL = rsU + (rsU - rd)(1 - T)(D/S).

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Notes About the New Propositions
1. When corporate taxes are added,
VL ≠ VU. VL increases as debt is
added to the capital structure, and
the greater the debt usage, the
higher the value of the firm.
2. rsL increases with leverage at a
slower rate when corporate taxes
are considered.

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1. Find VU and VL.
Note: Represents a 40% decline from the no
taxes situation.
VL = VU + TD = $2,142,857 + 0.4($1,000,000)
= $2,142,857 + $400,000
= $2,542,857.
VU = = = $2,142,857.
EBIT(1 - T)
rsU
$500,000(0.6)
0.14

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VL = D + S = $2,542,857
$2,542,857 = $1,000,000 + S
S = $1,542,857.
2. Find market value of Firm
L’s debt and equity.

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3. Find rsL.
rsL = rsU + (rsU - rd)(1 - T)(D/S)
= 14.0% + (14.0% - 8.0%)(0.6)( )
= 14.0% + 2.33% = 16.33%.
$1,000,000
$1,542,857

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4. Find Firm L’s WACC.
WACCL= (D/V)rd(1 - T) + (S/V)rs
= ( )(8.0%)(0.6)
+( )(16.33%)
= 1.89% + 9.91% = 11.80%.
When corporate taxes are considered, the
WACC is lower for L than for U.
$1,000,000
$2,542,857
$1,542,857
$2,542,857

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Cost of
Capital (%)
26
20
14
8
0 20 40 60 80 100
Debt/Value
Ratio (%)
MM relationship between capital costs
and leverage when corporate taxes are
considered.
rs
WACC
rd(1 - T)

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Value of Firm, V (%)
4
3
2
1
0 0.5 1.0 1.5 2.0 2.5
Debt
(Millions of $)
VL
VU
MM relationship between value and debt
when corporate taxes are considered.
Under MM with corporate taxes, the firm’s value
increases continuously as more and more debt is used.
TD

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Assume investors have the following
tax rates: Td = 30% and Ts = 12%. What
is the gain from leverage according to
the Miller model?
Miller’s Proposition I:
VL = VU + [1 - ]D.
Tc = corporate tax rate.
Td = personal tax rate on debt income.
Ts = personal tax rate on stock income.
(1 - Tc)(1 - Ts)
(1 - Td)

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Tc = 40%, Td = 30%, and Ts = 12%.
VL = VU + [1 - ]D
= VU + (1 - 0.75)D
= VU + 0.25D.
Value rises with debt; each $100 increase
in debt raises L’s value by $25.
(1 - 0.40)(1 - 0.12)
(1 - 0.30)

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How does this gain compare to the gain
in the MM model with corporate taxes?
If only corporate taxes, then
VL = VU + TcD = VU + 0.40D.
Here $100 of debt raises value by
$40. Thus, personal taxes lowers the
gain from leverage, but the net effect
depends on tax rates.
(More...)

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If Ts declines, while Tc and Td remain
constant, the slope coefficient
(which shows the benefit of debt) is
decreased.
A company with a low payout ratio
gets lower benefits under the Miller
model than a company with a high
payout, because a low payout
decreases Ts.

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When Miller brought in personal
taxes, the value enhancement of debt
was lowered. Why?
1. Corporate tax laws favor debt over
equity financing because interest
expense is tax deductible while
dividends are not.
(More...)

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2. However, personal tax laws favor
equity over debt because stocks
provide both tax deferral and a
lower capital gains tax rate.
3. This lowers the relative cost of
equity vis-a-vis MM’s no-personal-
tax world and decreases the spread
between debt and equity costs.
4. Thus, some of the advantage of debt
financing is lost, so debt financing
is less valuable to firms.

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What does capital structure theory
prescribe for corporate managers?
1. MM, No Taxes: Capital structure is
irrelevant--no impact on value or WACC.
2. MM, Corporate Taxes: Value increases,
so firms should use (almost) 100% debt
financing.
3. Miller, Personal Taxes: Value increases,
but less than under MM, so again firms
should use (almost) 100% debt financing.

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1. Firms don’t follow MM/Miller to 100%
debt. Debt ratios average about 40%.
2. However, debt ratios did increase after
MM. Many think debt ratios were too
low, and MM led to changes in financial
policies.
Do firms follow the recommendations
of capital structure theory?

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How is all of this analysis different if
firms U and L are growing?
Under MM (with taxes and no growth)
VL = VU + TD
This assumes the tax shield is
discounted at the cost of debt.
Assume the growth rate is 7%
The debt tax shield will be larger if
the firms grow:

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7% growth, TS discount rate of rTS
Value of (growing) tax shield =
VTS = rdTD/(rTS –g)
So value of levered firm =
VL = VU + rdTD/(rTS – g)

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What should rTS be?
The smaller is rTS, the larger the value
of the tax shield. If rTS < rsU, then with
rapid growth the tax shield becomes
unrealistically large—rTS must be
equal to rU to give reasonable results
when there is growth. So we assume
rTS = rsU.

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Levered cost of equity
In this case, the levered cost of
equity is rsL = rsU + (rsU – rd)(D/S)
This looks just like MM without taxes
even though we allow taxes and
allow for growth. The reason is if rTS
= rsU, then larger values of the tax
shield don't change the risk of the
equity.

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Levered beta
If there is growth and rTS = rsU then the
equation that is equivalent to the
Hamada equation is
βL = βU + (βU - βD)(D/S)
Notice: This looks like Hamada
without taxes. Again, this is because
in this case the tax shield doesn't
change the risk of the equity.

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Relevant information for valuation
EBIT = $500,000
T = 40%
rU = 14% = rTS
rd = 8%
Required reinvestment in net
operating assets = 10% of EBIT =
$50,000.
Debt = $1,000,000

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Calculating VU
NOPAT = EBIT(1-T)
= $500,000 (.60) = $300,000
Investment in net op. assets
= EBIT (0.10) = $50,000
FCF = NOPAT – Inv. in net op. assets
= $300,000 - $50,000
= $250,000 (this is expected FCF
next year)

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Value of unlevered firm, VU
Value of unlevered firm =
VU = FCF/(rsU – g)
= $250,000/(0.14 – 0.07)
= $3,571,429

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Value of tax shield, VTS and VL
VTS = rdTD/(rsU –g)
= 0.08(0.40)$1,000,000/(0.14-0.07)
= $457,143
VL = VU + VTS
= $3,571,429 + $457,143
= $4,028,571

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Cost of equity and WACC
Just like with MM with taxes, the cost
of equity increases with D/V, and the
WACC declines.
But since rsL doesn't have the (1-T)
factor in it, for a given D/V, rsL is
greater than MM would predict, and
WACC is greater than MM would
predict.

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Costs of capital for MM and Extension
0.00%
5.00%
10.00%
15.00%
20.00%
25.00%
30.00%
35.00%
40.00%
0% 20% 40% 60% 80% 100%
D/V
CostofCapital
MM rsL
MM WACC
Extension rsL
Extension WACC

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What if L's debt is risky?
If L's debt is risky then, by definition,
management might default on it. The
decision to make a payment on the
debt or to default looks very much
like the decision whether to exercise
a call option. So the equity looks like
an option.

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Equity as an option
Suppose the firm has $2 million face value
of 1-year zero coupon debt, and the
current value of the firm (debt plus equity)
is $4 million.
If the firm pays off the debt when it matures,
the equity holders get to keep the firm. If
not, they get nothing because the
debtholders foreclose.

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Equity as an option
The equity holder's position looks like
a call option with
P = underlying value of firm = $4
million
X = exercise price = $2 million
t = time to maturity = 1 year
Suppose rRF = 6%
σ = volatility of debt + equity = 0.60

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Use Black-Scholes to price this option
V = P[N(d1)] - Xe-rRFt
[N(d2)].
d1 = .
σ t
d2 = d1 - σ t.
ln(P/X) + [rRF + (σ2
/2)]t

46. 17 - 46
Black-Scholes Solution
V = $4[N(d1)] - $2e-(0.06)(1.0)
[N(d2)].
ln($4/$2) + [(0.06 + 0.36/2)](1.0)
(0.60)(1.0)
= 1.5552.
d2 = d1 - (0.60)(1.0) = d1 - 0.60
= 1.5552 - 0.6000 = 0.9552.
d1 =

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N(d1) = N(1.5552) = 0.9401
N(d2) = N(0.9552) = 0.8383
Note: Values obtained from Excel using
NORMSDIST function.
V = $4(0.9401) - $2e-0.06
(0.8303)
= $3.7604 - $2(0.9418)(0.8303)
= $2.196 Million = Value of Equity

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Value of Debt
The value of debt must be what is left
over:
Value of debt = Total Value – Equity
= $4 million – 2.196 million
= $1.804 million

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This value of debt gives us a yield
Debt yield for 1-year zero coupon debt
= (face value / price) – 1
= ($2 million/ 1.804 million) – 1
= 10.9%

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How does σ affect an option's value?
Higher volatility σ means higher option
value.

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Managerial Incentives
When an investor buys a stock option,
the riskiness of the stock (σ) is
already determined. But a manager
can change a firm's σ by changing
the assets the firm invests in. That
means changing σ can change the
value of the equity, even if it doesn't
change the expected cash flows:

52. 17 - 52
Managerial Incentives
So changing σ can transfer wealth
from bondholders to stockholders by
making the option value of the stock
worth more, which makes what is
left, the debt value, worth less.

53. 17 - 53
Values of Debt and Equity for Different Volatilities
0.00
0.50
1.00
1.50
2.00
2.50
3.00
0.00 0.20 0.40 0.60 0.80 1.00
Volatility (sigma)
Value(millions)
Equity
Debt

54. 17 - 54
Bait and Switch
Managers who know this might tell
debtholders they are going to invest
in one kind of asset, and, instead,
invest in riskier assets. This is
called bait and switch and
bondholders will require higher
interest rates for firms that do this,
or refuse to do business with them.

55. 17 - 55
If the debt is risky coupon debt
If the risky debt has coupons, then
with each coupon payment
management has an option on an
option—if it makes the interest
payment then it purchases the right
to later make the principal payment
and keep the firm. This is called a
compound option.